{"history":[{"id":"0caabb7f-df6d-45bb-8547-f0d74c224997","wordChanged":"[aleph-null](https://bliptext.com/articles/aleph-number)","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a ****cardinal** number**, or **cardinal** for short, is what is commonly called the number of elements of a [set](https://bliptext.com/articles/set-mathematics).\nIn the case of a [finite set](https://bliptext.com/articles/finite-set), its cardinal number, or [cardinality](https://bliptext.com/articles/cardinality) is therefore a [natural number](https://bliptext.com/articles/natural-number).\nFor dealing with the case of [infinite sets](https://bliptext.com/articles/infinite-set), the [infinite cardinal numbers](https://bliptext.com/articles/transfinite-number) have been introduced, which are often denoted with the [Hebrew letter](https://bliptext.com/articles/hebrew-alphabet) \\[aleph](https://bliptext.com/articles/aleph-hebrew) ([aleph](https://bliptext.com/articles/aleph-hebrew)) marked with subscript indicating their rank among the infinite cardinals.\nCardinality is defined in terms of [bijective functions](https://bliptext.com/articles/bijective-function).\nTwo sets have the same cardinality [if, and only if](https://bliptext.com/articles/if-and-only-if), there is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) (bijection) between the elements of the two sets.\nIn the case of finite sets, this agrees with the intuitive notion of number of elements.\nIn the case of infinite sets, the behavior is more complex.\nA fundamental theorem due to [Georg Cantor](https://bliptext.com/articles/georg-cantor) shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [real numbers](https://bliptext.com/articles/real-number) is greater than the cardinality of the set of natural numbers.\nIt is also possible for a [proper subset](https://bliptext.com/articles/proper-subset) of an infinite set to have the same cardinality as the original set—something that cannot happen with [proper subset](https://bliptext.com/articles/proper-subset)s of finite sets.\nThere is a [transfinite sequence](https://bliptext.com/articles/transfinite-sequence) of cardinal numbers:\nThis sequence starts with the [natural numbers](https://bliptext.com/articles/natural-number) including zero (finite cardinals), which are followed by the [aleph numbers](https://bliptext.com/articles/aleph-number).\nThe aleph numbers are indexed by [ordinal numbers](https://bliptext.com/articles/ordinal-number).\nIf the [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is true, this transfinite sequence includes every cardinal number.\nIf the axiom of choice is not true (see ), there are infinite cardinals that are not aleph numbers.\n[Cardinality](https://bliptext.com/articles/cardinality) is studied for its own sake as part of [set theory](https://bliptext.com/articles/set-theory).\nIt is also a tool used in branches of mathematics including [model theory](https://bliptext.com/articles/model-theory), [combinatorics](https://bliptext.com/articles/combinatorics), [abstract algebra](https://bliptext.com/articles/abstract-algebra) and [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIn [category theory](https://bliptext.com/articles/category-theory), the cardinal numbers form a [skeleton](https://bliptext.com/articles/skeleton-category-theory) of the [category of sets](https://bliptext.com/articles/category-of-sets).\n# [A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. | upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg]\n# [Aleph-null, the smallest infinite cardinal | upload.wikimedia.org/wikipedia/commons/e/e0/Aleph0.svg]\n\n# History\nThe notion of cardinality, as now understood, was formulated by [Georg Cantor](https://bliptext.com/articles/georg-cantor), the originator of [set theory](https://bliptext.com/articles/set-theory), in 1874–1884.\nCardinality can be used to compare an aspect of finite sets.\nFor example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three.\nThis is established by the existence of a [bijection](https://bliptext.com/articles/bijection) (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.\nCantor applied his concept of bijection to infinite sets (for example the set of natural numbers **N** = {0, 1, 2, 3, ...}).\nThus, he called all sets having a bijection with **N** denumerable (countably infinite) sets, which all share the same cardinal number.\nThis cardinal number is called 4 null.\nHe called the cardinal numbers of infinite sets [transfinite cardinal numbers](https://bliptext.com/articles/transfinite-cardinal-numbers).\nCantor proved that any [unbounded subset](https://bliptext.com/articles/bounded-set) of ****N**** has the same cardinality as ****N****, even though this might appear to run contrary to intuition.\nHe also proved that the set of all [ordered pairs](https://bliptext.com/articles/ordered-pair) of natural numbers is denumerable; this implies that the set of all [rational numbers](https://bliptext.com/articles/rational-number) is also denumerable, since every rational can be represented by a pair of integers.\nHe later proved that the set of all real [algebraic numbers](https://bliptext.com/articles/algebraic-number) is also denumerable.\nEach real algebraic number z may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai ∈ **Z** together with a pair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies in the interval (b0, b1).\nIn his 1874 paper \"On a Property of the Collection of All Real Algebraic **N**umbers\", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of **N**.\nHis proof used an argument with [nested intervals](https://bliptext.com/articles/nested-intervals), but in an 1891 paper, he proved the same result using his ingenious and much simpler [diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument).\nThe new cardinal number of the set of real numbers is called the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) and Cantor used the symbol for it.\nCantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (\\aleph_0, aleph-null), and that for every cardinal number there is a next-larger cardinal\nHis [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) is the proposition that the cardinality of the set of real numbers is the same as \\aleph_1.\nThis hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them.\nThis was shown in 1963 by [Paul Cohen](https://bliptext.com/articles/paul-cohen-mathematician), complementing earlier work by [Kurt Gödel](https://bliptext.com/articles/kurt-g-del) in 1940.\n\n# Motivation\nIn informal use, a cardinal number is what is normally referred to as a [counting number](https://bliptext.com/articles/counting-number), provided that 0 is included: 0, 1, 2, .... They may be identified with the [natural numbers](https://bliptext.com/articles/natural-numbers) beginning with 0.\nThe counting numbers are exactly what can be defined formally as the [finite](https://bliptext.com/articles/finite-set) cardinal numbers.\nInfinite cardinals only occur in higher-level mathematics and [logic](https://bliptext.com/articles/logic).\nMore formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence.\nFor finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size.\nFor example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.\nHowever, when dealing with [infinite sets](https://bliptext.com/articles/infinite-set), it is essential to distinguish between the two, since the two notions are in fact different for [infinite sets](https://bliptext.com/articles/infinite-set).\nConsidering the position aspect leads to [ordinal numbers](https://bliptext.com/articles/ordinal-numbers), while the size aspect is generalized by the cardinal numbers described here.\nThe intuition behind the formal definition of cardinal is the construction of a notion of the relative size or \"bigness\" of a set, without reference to the kind of members which it has.\nFor finite sets this is easy; one simply counts the number of elements a set has.\nIn order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.\nA set Y is at least as big as a set X if there is an [injective](https://bliptext.com/articles/injective-function) [mapping](https://bliptext.com/articles/map-mathematics) from the elements of X to the elements of Y.\nAn injective mapping identifies each element of the set X with a unique element of the set Y.\nThis is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:\nwhich is injective, and hence conclude that Y has cardinality greater than or equal to X.\nThe element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a [bijective](https://bliptext.com/articles/bijective) mapping.\nThe advantage of this notion is that it can be extended to infinite sets.\nWe can then extend this to an equality-style relation.\nTwo [sets](https://bliptext.com/articles/set-mathematics) X and Y are said to have the same cardinality if there exists a [bijection](https://bliptext.com/articles/bijection) between X and Y.\nBy the [Schroeder–Bernstein theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem), this is equivalent to there being both an injective mapping from X to Y, and an injective mapping from Y to X.\nWe then write |X| = |Y|.\nThe cardinal number of X itself is often defined as the least ordinal a with |a| = |X|.\nThis is called the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment); for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the [well-ordering principle](https://bliptext.com/articles/well-ordering-principle).\nIt is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.\nThe classic example used is that of the infinite hotel paradox, also called [Hilbert's paradox of the Grand Hotel](https://bliptext.com/articles/hilbert-s-paradox-of-the-grand-hotel).\nSupposing there is an innkeeper at a hotel with an infinite number of rooms.\nThe hotel is full, and then a new guest arrives.\nIt is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant.\nWe can explicitly write a segment of this mapping:\nWith this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown.\nThis motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a [Dedekind-infinite set](https://bliptext.com/articles/dedekind-infinite-set)); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.\nWhen considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets.\nIt happens that it does not; by considering the above example we can see that if some object \"one greater than infinity\" exists, then it must have the same cardinality as the infinite set we started out with.\nIt is possible to use a different formal notion for number, called [ordinals](https://bliptext.com/articles/ordinal-number), based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.\nIt can be proved that the cardinality of the [real numbers](https://bliptext.com/articles/real-number) is greater than that of the natural numbers just described.\nThis can be visualized using [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument); classic questions of cardinality (for instance the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis)) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.\nIn more recent times, mathematicians have been describing the properties of larger and larger cardinals.\nSince cardinality is such a common concept in mathematics, a variety of names are in use.\nSameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.\nIt is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.\n\n# Formal definition\nFormally, assuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), the cardinality of a set X is the least [ordinal number](https://bliptext.com/articles/ordinal-number) α such that there is a bijection between X and α.\nThis definition is known as the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment).\nIf the axiom of choice is not assumed, then a different approach is needed.\nThe oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and [Principia Mathematica](https://bliptext.com/articles/principia-mathematica)) is as the class [X] of all sets that are equinumerous with X.\nThis does not work in [ZFC](https://bliptext.com/articles/zfc) or other related systems of [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory) because if X is non-empty, this collection is too large to be a set.\nIn fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X, and so by the [axiom of limitation of size](https://bliptext.com/articles/axiom-of-limitation-of-size), [X] is a proper class.\nThe definition does work however in [type theory](https://bliptext.com/articles/type-theory) and in [New Foundations](https://bliptext.com/articles/new-foundations) and related systems.\nHowever, if we restrict from this class to those equinumerous with X that have the least [rank](https://bliptext.com/articles/rank-set-theory), then it will work (this is a trick due to [Dana Scott](https://bliptext.com/articles/dana-scott): it works because the collection of objects with any given [rank](https://bliptext.com/articles/rank-set-theory) is a set).\nVon Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers.\nHowever, they differ for infinite numbers.\nFor example, in ordinal arithmetic while in cardinal arithmetic, although the von Neumann assignment puts.\nOn the other hand, Scott's trick implies that the cardinal number 0 is, which is also the ordinal number 1, and this may be confusing.\nA possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.\nFormally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an [injective](https://bliptext.com/articles/injective) function from X to Y.\nThe [Cantor–Bernstein–Schroeder theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem) states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|.\nThe axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|.\nA set X is [Dedekind-infinite](https://bliptext.com/articles/dedekind-infinite) if there exists a [proper subset](https://bliptext.com/articles/proper-subset) Y of X with |X| = |Y|, and [Dedekind-finite](https://bliptext.com/articles/dedekind-finite) if such a subset does not exist.\nThe [finite](https://bliptext.com/articles/finite-set) cardinals are just the [natural numbers](https://bliptext.com/articles/natural-numbers), in the sense that a set X is [finite](https://bliptext.com/articles/finite-set) if and only if |X| = |n| = n for some natural number n.\nAny other set is [infinite](https://bliptext.com/articles/infinite-set).\nAssuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones.\nIt can also be proved that the cardinal \\aleph_0 ([aleph null](https://bliptext.com/articles/aleph-null) or aleph-0, where aleph is the first letter in the [Hebrew alphabet](https://bliptext.com/articles/hebrew-alphabet), represented \\aleph) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality \\aleph_0).\nThe next larger cardinal is denoted by \\aleph_1, and so on.\nFor every ordinal α, there is a cardinal number and this list exhausts all infinite cardinal numbers.\n\n# Cardinal arithmetic\nWe can define [arithmetic](https://bliptext.com/articles/arithmetic) operations on cardinal numbers that generalize the ordinary operations for natural numbers.\nIt can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers.\nFurthermore, these operations share many properties with ordinary arithmetic.\n\n## Successor cardinal\nIf the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor.\n(Without the axiom of choice, using [Hartogs' theorem](https://bliptext.com/articles/hartogs-number), it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that ) For finite cardinals, the successor is simply κ + 1.\nFor infinite cardinals, the successor cardinal differs from the [successor ordinal](https://bliptext.com/articles/successor-ordinal).\n\n## Cardinal addition\nIf X and Y are [disjoint](https://bliptext.com/articles/disjoint-sets), addition is given by the [union](https://bliptext.com/articles/union-set-theory) of X and Y.\nIf the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).\nZero is an additive identity κ + 0 = 0 + κ = κ.\nAddition is [associative](https://bliptext.com/articles/associative) (κ + μ) + ν = κ + (μ + ν).\nAddition is [commutative](https://bliptext.com/articles/commutative) κ + μ = μ + κ.\nAddition is non-decreasing in both arguments:\nAssuming the axiom of choice, addition of infinite cardinal numbers is easy.\nIf either κ or μ is infinite, then\n\n### Subtraction\nAssuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ.\nIt will be unique (and equal to σ) if and only if μ < σ.\n\n## Cardinal multiplication\nThe product of cardinals comes from the [Cartesian product](https://bliptext.com/articles/cartesian-product).\nκ·0 = 0·κ = 0.\nκ·μ = 0 → (κ = 0 or μ = 0).\nOne is a multiplicative identity κ·1 = 1·κ = κ.\nMultiplication is associative (κ·μ)·ν = κ·(μ·ν).\nMultiplication is [commutative](https://bliptext.com/articles/commutative) κ·μ = μ·κ.\nMultiplication is non-decreasing in both arguments:\nκ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).\nMultiplication [distributes](https://bliptext.com/articles/distributivity) over addition:\nκ·(μ + ν) = κ·μ + κ·ν and\n(μ + ν)·κ = μ·κ + ν·κ.\nAssuming the axiom of choice, multiplication of infinite cardinal numbers is also easy.\nIf either κ or μ is infinite and both are non-zero, then\n\n### Division\nAssuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π.\nIt will be unique (and equal to π) if and only if μ < π.\n\n## Cardinal exponentiation\nExponentiation is given by where XY is the set of all [functions](https://bliptext.com/articles/function-mathematics) from Y to X.\nIt is easy to check that the right-hand side depends only on {|X|} and {|Y|}.\nExponentiation is non-decreasing in both arguments:\n2undefined is the cardinality of the [power set](https://bliptext.com/articles/power-set) of the set X and [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument) shows that 2undefined > |X| for any set X.\nThis proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ).\nIn fact, the [class](https://bliptext.com/articles/class-set-theory) of cardinals is a proper [class](https://bliptext.com/articles/class-set-theory).\n(This proof fails in some set theories, notably [New Foundations](https://bliptext.com/articles/new-foundations).)\nAll the remaining propositions in this section assume the axiom of choice:\nIf 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:\nUsing [König's theorem](https://bliptext.com/articles/k-nig-s-theorem-set-theory), one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the [cofinality](https://bliptext.com/articles/cofinality) of κ.\n\n### Roots\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying will be \\kappa.\n\n### Logarithms\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying.\nHowever, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy.\nThe logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ.\nLogarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of [cardinal invariants](https://bliptext.com/articles/cardinal-invariant) of [topological spaces](https://bliptext.com/articles/topological-space), though they lack some of the properties that logarithms of positive real numbers possess.\n\n# The continuum hypothesis\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) (CH) states that there are no cardinals strictly between \\aleph_0 and The latter cardinal number is also often denoted by ; it is the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) (the set of [real numbers](https://bliptext.com/articles/real-number)).\nIn this case\nSimilarly, the [generalized continuum hypothesis](https://bliptext.com/articles/generalized-continuum-hypothesis) (GCH) states that for every infinite cardinal \\kappa, there are no cardinals strictly between \\kappa and 2^\\kappa.\nBoth the continuum hypothesis and the generalized continuum hypothesis have been proved to be independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ([ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory)).\nIndeed, [Easton's theorem](https://bliptext.com/articles/easton-s-theorem) shows that, for [regular cardinals](https://bliptext.com/articles/regular-cardinal) \\kappa, the only restrictions ZFC places on the cardinality of 2^\\kappa are that, and that the exponential function is non-decreasing.","wordIndex":470,"createdAt":"2026-01-28T10:23:23.759Z","user":{"id":"mfvswiH9fwHI8f36yNwHR40968SZeIKD","name":"den3den42069","image":"https://cdn.discordapp.com/avatars/975452809707278427/5e0588882214f34dc5f3b36f7e9c9ff5.png"}},{"id":"c5c96486-5fe2-4cd5-8c8e-83762db1d739","wordChanged":"\\aleph_0,","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a ****cardinal** number**, or **cardinal** for short, is what is commonly called the number of elements of a [set](https://bliptext.com/articles/set-mathematics).\nIn the case of a [finite set](https://bliptext.com/articles/finite-set), its cardinal number, or [cardinality](https://bliptext.com/articles/cardinality) is therefore a [natural number](https://bliptext.com/articles/natural-number).\nFor dealing with the case of [infinite sets](https://bliptext.com/articles/infinite-set), the [infinite cardinal numbers](https://bliptext.com/articles/transfinite-number) have been introduced, which are often denoted with the [Hebrew letter](https://bliptext.com/articles/hebrew-alphabet) \\[aleph](https://bliptext.com/articles/aleph-hebrew) ([aleph](https://bliptext.com/articles/aleph-hebrew)) marked with subscript indicating their rank among the infinite cardinals.\nCardinality is defined in terms of [bijective functions](https://bliptext.com/articles/bijective-function).\nTwo sets have the same cardinality [if, and only if](https://bliptext.com/articles/if-and-only-if), there is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) (bijection) between the elements of the two sets.\nIn the case of finite sets, this agrees with the intuitive notion of number of elements.\nIn the case of infinite sets, the behavior is more complex.\nA fundamental theorem due to [Georg Cantor](https://bliptext.com/articles/georg-cantor) shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [real numbers](https://bliptext.com/articles/real-number) is greater than the cardinality of the set of natural numbers.\nIt is also possible for a [proper subset](https://bliptext.com/articles/proper-subset) of an infinite set to have the same cardinality as the original set—something that cannot happen with [proper subset](https://bliptext.com/articles/proper-subset)s of finite sets.\nThere is a [transfinite sequence](https://bliptext.com/articles/transfinite-sequence) of cardinal numbers:\nThis sequence starts with the [natural numbers](https://bliptext.com/articles/natural-number) including zero (finite cardinals), which are followed by the [aleph numbers](https://bliptext.com/articles/aleph-number).\nThe aleph numbers are indexed by [ordinal numbers](https://bliptext.com/articles/ordinal-number).\nIf the [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is true, this transfinite sequence includes every cardinal number.\nIf the axiom of choice is not true (see ), there are infinite cardinals that are not aleph numbers.\n[Cardinality](https://bliptext.com/articles/cardinality) is studied for its own sake as part of [set theory](https://bliptext.com/articles/set-theory).\nIt is also a tool used in branches of mathematics including [model theory](https://bliptext.com/articles/model-theory), [combinatorics](https://bliptext.com/articles/combinatorics), [abstract algebra](https://bliptext.com/articles/abstract-algebra) and [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIn [category theory](https://bliptext.com/articles/category-theory), the cardinal numbers form a [skeleton](https://bliptext.com/articles/skeleton-category-theory) of the [category of sets](https://bliptext.com/articles/category-of-sets).\n# [A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. | upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg]\n# [Aleph-null, the smallest infinite cardinal | upload.wikimedia.org/wikipedia/commons/e/e0/Aleph0.svg]\n\n# History\nThe notion of cardinality, as now understood, was formulated by [Georg Cantor](https://bliptext.com/articles/georg-cantor), the originator of [set theory](https://bliptext.com/articles/set-theory), in 1874–1884.\nCardinality can be used to compare an aspect of finite sets.\nFor example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three.\nThis is established by the existence of a [bijection](https://bliptext.com/articles/bijection) (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.\nCantor applied his concept of bijection to infinite sets (for example the set of natural numbers **N** = {0, 1, 2, 3, ...}).\nThus, he called all sets having a bijection with **N** denumerable (countably infinite) sets, which all share the same cardinal number.\nThis cardinal number is called 4 [aleph-null](https://bliptext.com/articles/aleph-number).\nHe called the cardinal numbers of infinite sets [transfinite cardinal numbers](https://bliptext.com/articles/transfinite-cardinal-numbers).\nCantor proved that any [unbounded subset](https://bliptext.com/articles/bounded-set) of ****N**** has the same cardinality as ****N****, even though this might appear to run contrary to intuition.\nHe also proved that the set of all [ordered pairs](https://bliptext.com/articles/ordered-pair) of natural numbers is denumerable; this implies that the set of all [rational numbers](https://bliptext.com/articles/rational-number) is also denumerable, since every rational can be represented by a pair of integers.\nHe later proved that the set of all real [algebraic numbers](https://bliptext.com/articles/algebraic-number) is also denumerable.\nEach real algebraic number z may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai ∈ **Z** together with a pair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies in the interval (b0, b1).\nIn his 1874 paper \"On a Property of the Collection of All Real Algebraic **N**umbers\", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of **N**.\nHis proof used an argument with [nested intervals](https://bliptext.com/articles/nested-intervals), but in an 1891 paper, he proved the same result using his ingenious and much simpler [diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument).\nThe new cardinal number of the set of real numbers is called the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) and Cantor used the symbol for it.\nCantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (\\aleph_0, aleph-null), and that for every cardinal number there is a next-larger cardinal\nHis [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) is the proposition that the cardinality of the set of real numbers is the same as \\aleph_1.\nThis hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them.\nThis was shown in 1963 by [Paul Cohen](https://bliptext.com/articles/paul-cohen-mathematician), complementing earlier work by [Kurt Gödel](https://bliptext.com/articles/kurt-g-del) in 1940.\n\n# Motivation\nIn informal use, a cardinal number is what is normally referred to as a [counting number](https://bliptext.com/articles/counting-number), provided that 0 is included: 0, 1, 2, .... They may be identified with the [natural numbers](https://bliptext.com/articles/natural-numbers) beginning with 0.\nThe counting numbers are exactly what can be defined formally as the [finite](https://bliptext.com/articles/finite-set) cardinal numbers.\nInfinite cardinals only occur in higher-level mathematics and [logic](https://bliptext.com/articles/logic).\nMore formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence.\nFor finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size.\nFor example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.\nHowever, when dealing with [infinite sets](https://bliptext.com/articles/infinite-set), it is essential to distinguish between the two, since the two notions are in fact different for [infinite sets](https://bliptext.com/articles/infinite-set).\nConsidering the position aspect leads to [ordinal numbers](https://bliptext.com/articles/ordinal-numbers), while the size aspect is generalized by the cardinal numbers described here.\nThe intuition behind the formal definition of cardinal is the construction of a notion of the relative size or \"bigness\" of a set, without reference to the kind of members which it has.\nFor finite sets this is easy; one simply counts the number of elements a set has.\nIn order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.\nA set Y is at least as big as a set X if there is an [injective](https://bliptext.com/articles/injective-function) [mapping](https://bliptext.com/articles/map-mathematics) from the elements of X to the elements of Y.\nAn injective mapping identifies each element of the set X with a unique element of the set Y.\nThis is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:\nwhich is injective, and hence conclude that Y has cardinality greater than or equal to X.\nThe element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a [bijective](https://bliptext.com/articles/bijective) mapping.\nThe advantage of this notion is that it can be extended to infinite sets.\nWe can then extend this to an equality-style relation.\nTwo [sets](https://bliptext.com/articles/set-mathematics) X and Y are said to have the same cardinality if there exists a [bijection](https://bliptext.com/articles/bijection) between X and Y.\nBy the [Schroeder–Bernstein theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem), this is equivalent to there being both an injective mapping from X to Y, and an injective mapping from Y to X.\nWe then write |X| = |Y|.\nThe cardinal number of X itself is often defined as the least ordinal a with |a| = |X|.\nThis is called the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment); for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the [well-ordering principle](https://bliptext.com/articles/well-ordering-principle).\nIt is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.\nThe classic example used is that of the infinite hotel paradox, also called [Hilbert's paradox of the Grand Hotel](https://bliptext.com/articles/hilbert-s-paradox-of-the-grand-hotel).\nSupposing there is an innkeeper at a hotel with an infinite number of rooms.\nThe hotel is full, and then a new guest arrives.\nIt is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant.\nWe can explicitly write a segment of this mapping:\nWith this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown.\nThis motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a [Dedekind-infinite set](https://bliptext.com/articles/dedekind-infinite-set)); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.\nWhen considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets.\nIt happens that it does not; by considering the above example we can see that if some object \"one greater than infinity\" exists, then it must have the same cardinality as the infinite set we started out with.\nIt is possible to use a different formal notion for number, called [ordinals](https://bliptext.com/articles/ordinal-number), based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.\nIt can be proved that the cardinality of the [real numbers](https://bliptext.com/articles/real-number) is greater than that of the natural numbers just described.\nThis can be visualized using [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument); classic questions of cardinality (for instance the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis)) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.\nIn more recent times, mathematicians have been describing the properties of larger and larger cardinals.\nSince cardinality is such a common concept in mathematics, a variety of names are in use.\nSameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.\nIt is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.\n\n# Formal definition\nFormally, assuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), the cardinality of a set X is the least [ordinal number](https://bliptext.com/articles/ordinal-number) α such that there is a bijection between X and α.\nThis definition is known as the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment).\nIf the axiom of choice is not assumed, then a different approach is needed.\nThe oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and [Principia Mathematica](https://bliptext.com/articles/principia-mathematica)) is as the class [X] of all sets that are equinumerous with X.\nThis does not work in [ZFC](https://bliptext.com/articles/zfc) or other related systems of [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory) because if X is non-empty, this collection is too large to be a set.\nIn fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X, and so by the [axiom of limitation of size](https://bliptext.com/articles/axiom-of-limitation-of-size), [X] is a proper class.\nThe definition does work however in [type theory](https://bliptext.com/articles/type-theory) and in [New Foundations](https://bliptext.com/articles/new-foundations) and related systems.\nHowever, if we restrict from this class to those equinumerous with X that have the least [rank](https://bliptext.com/articles/rank-set-theory), then it will work (this is a trick due to [Dana Scott](https://bliptext.com/articles/dana-scott): it works because the collection of objects with any given [rank](https://bliptext.com/articles/rank-set-theory) is a set).\nVon Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers.\nHowever, they differ for infinite numbers.\nFor example, in ordinal arithmetic while in cardinal arithmetic, although the von Neumann assignment puts.\nOn the other hand, Scott's trick implies that the cardinal number 0 is, which is also the ordinal number 1, and this may be confusing.\nA possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.\nFormally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an [injective](https://bliptext.com/articles/injective) function from X to Y.\nThe [Cantor–Bernstein–Schroeder theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem) states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|.\nThe axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|.\nA set X is [Dedekind-infinite](https://bliptext.com/articles/dedekind-infinite) if there exists a [proper subset](https://bliptext.com/articles/proper-subset) Y of X with |X| = |Y|, and [Dedekind-finite](https://bliptext.com/articles/dedekind-finite) if such a subset does not exist.\nThe [finite](https://bliptext.com/articles/finite-set) cardinals are just the [natural numbers](https://bliptext.com/articles/natural-numbers), in the sense that a set X is [finite](https://bliptext.com/articles/finite-set) if and only if |X| = |n| = n for some natural number n.\nAny other set is [infinite](https://bliptext.com/articles/infinite-set).\nAssuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones.\nIt can also be proved that the cardinal \\aleph_0 ([aleph null](https://bliptext.com/articles/aleph-null) or aleph-0, where aleph is the first letter in the [Hebrew alphabet](https://bliptext.com/articles/hebrew-alphabet), represented \\aleph) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality \\aleph_0).\nThe next larger cardinal is denoted by \\aleph_1, and so on.\nFor every ordinal α, there is a cardinal number and this list exhausts all infinite cardinal numbers.\n\n# Cardinal arithmetic\nWe can define [arithmetic](https://bliptext.com/articles/arithmetic) operations on cardinal numbers that generalize the ordinary operations for natural numbers.\nIt can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers.\nFurthermore, these operations share many properties with ordinary arithmetic.\n\n## Successor cardinal\nIf the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor.\n(Without the axiom of choice, using [Hartogs' theorem](https://bliptext.com/articles/hartogs-number), it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that ) For finite cardinals, the successor is simply κ + 1.\nFor infinite cardinals, the successor cardinal differs from the [successor ordinal](https://bliptext.com/articles/successor-ordinal).\n\n## Cardinal addition\nIf X and Y are [disjoint](https://bliptext.com/articles/disjoint-sets), addition is given by the [union](https://bliptext.com/articles/union-set-theory) of X and Y.\nIf the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).\nZero is an additive identity κ + 0 = 0 + κ = κ.\nAddition is [associative](https://bliptext.com/articles/associative) (κ + μ) + ν = κ + (μ + ν).\nAddition is [commutative](https://bliptext.com/articles/commutative) κ + μ = μ + κ.\nAddition is non-decreasing in both arguments:\nAssuming the axiom of choice, addition of infinite cardinal numbers is easy.\nIf either κ or μ is infinite, then\n\n### Subtraction\nAssuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ.\nIt will be unique (and equal to σ) if and only if μ < σ.\n\n## Cardinal multiplication\nThe product of cardinals comes from the [Cartesian product](https://bliptext.com/articles/cartesian-product).\nκ·0 = 0·κ = 0.\nκ·μ = 0 → (κ = 0 or μ = 0).\nOne is a multiplicative identity κ·1 = 1·κ = κ.\nMultiplication is associative (κ·μ)·ν = κ·(μ·ν).\nMultiplication is [commutative](https://bliptext.com/articles/commutative) κ·μ = μ·κ.\nMultiplication is non-decreasing in both arguments:\nκ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).\nMultiplication [distributes](https://bliptext.com/articles/distributivity) over addition:\nκ·(μ + ν) = κ·μ + κ·ν and\n(μ + ν)·κ = μ·κ + ν·κ.\nAssuming the axiom of choice, multiplication of infinite cardinal numbers is also easy.\nIf either κ or μ is infinite and both are non-zero, then\n\n### Division\nAssuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π.\nIt will be unique (and equal to π) if and only if μ < π.\n\n## Cardinal exponentiation\nExponentiation is given by where XY is the set of all [functions](https://bliptext.com/articles/function-mathematics) from Y to X.\nIt is easy to check that the right-hand side depends only on {|X|} and {|Y|}.\nExponentiation is non-decreasing in both arguments:\n2undefined is the cardinality of the [power set](https://bliptext.com/articles/power-set) of the set X and [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument) shows that 2undefined > |X| for any set X.\nThis proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ).\nIn fact, the [class](https://bliptext.com/articles/class-set-theory) of cardinals is a proper [class](https://bliptext.com/articles/class-set-theory).\n(This proof fails in some set theories, notably [New Foundations](https://bliptext.com/articles/new-foundations).)\nAll the remaining propositions in this section assume the axiom of choice:\nIf 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:\nUsing [König's theorem](https://bliptext.com/articles/k-nig-s-theorem-set-theory), one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the [cofinality](https://bliptext.com/articles/cofinality) of κ.\n\n### Roots\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying will be \\kappa.\n\n### Logarithms\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying.\nHowever, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy.\nThe logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ.\nLogarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of [cardinal invariants](https://bliptext.com/articles/cardinal-invariant) of [topological spaces](https://bliptext.com/articles/topological-space), though they lack some of the properties that logarithms of positive real numbers possess.\n\n# The continuum hypothesis\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) (CH) states that there are no cardinals strictly between \\aleph_0 and The latter cardinal number is also often denoted by ; it is the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) (the set of [real numbers](https://bliptext.com/articles/real-number)).\nIn this case\nSimilarly, the [generalized continuum hypothesis](https://bliptext.com/articles/generalized-continuum-hypothesis) (GCH) states that for every infinite cardinal \\kappa, there are no cardinals strictly between \\kappa and 2^\\kappa.\nBoth the continuum hypothesis and the generalized continuum hypothesis have been proved to be independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ([ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory)).\nIndeed, [Easton's theorem](https://bliptext.com/articles/easton-s-theorem) shows that, for [regular cardinals](https://bliptext.com/articles/regular-cardinal) \\kappa, the only restrictions ZFC places on the cardinality of 2^\\kappa are that, and that the exponential function is non-decreasing.","wordIndex":469,"createdAt":"2026-01-28T10:22:48.634Z","user":{"id":"mfvswiH9fwHI8f36yNwHR40968SZeIKD","name":"den3den42069","image":"https://cdn.discordapp.com/avatars/975452809707278427/5e0588882214f34dc5f3b36f7e9c9ff5.png"}}],"article":{"id":"ffced29f-7abf-428f-b75a-b16151421afb","title":"Cardinal number","slug":"cardinal-number","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a ****cardinal** number**, or **cardinal** for short, is what is commonly called the number of elements of a [set](https://bliptext.com/articles/set-mathematics).\nIn the case of a [finite set](https://bliptext.com/articles/finite-set), its cardinal number, or [cardinality](https://bliptext.com/articles/cardinality) is therefore a [natural number](https://bliptext.com/articles/natural-number).\nFor dealing with the case of [infinite sets](https://bliptext.com/articles/infinite-set), the [infinite cardinal numbers](https://bliptext.com/articles/transfinite-number) have been introduced, which are often denoted with the [Hebrew letter](https://bliptext.com/articles/hebrew-alphabet) \\[aleph](https://bliptext.com/articles/aleph-hebrew) ([aleph](https://bliptext.com/articles/aleph-hebrew)) marked with subscript indicating their rank among the infinite cardinals.\nCardinality is defined in terms of [bijective functions](https://bliptext.com/articles/bijective-function).\nTwo sets have the same cardinality [if, and only if](https://bliptext.com/articles/if-and-only-if), there is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) (bijection) between the elements of the two sets.\nIn the case of finite sets, this agrees with the intuitive notion of number of elements.\nIn the case of infinite sets, the behavior is more complex.\nA fundamental theorem due to [Georg Cantor](https://bliptext.com/articles/georg-cantor) shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [real numbers](https://bliptext.com/articles/real-number) is greater than the cardinality of the set of natural numbers.\nIt is also possible for a [proper subset](https://bliptext.com/articles/proper-subset) of an infinite set to have the same cardinality as the original set—something that cannot happen with [proper subset](https://bliptext.com/articles/proper-subset)s of finite sets.\nThere is a [transfinite sequence](https://bliptext.com/articles/transfinite-sequence) of cardinal numbers:\nThis sequence starts with the [natural numbers](https://bliptext.com/articles/natural-number) including zero (finite cardinals), which are followed by the [aleph numbers](https://bliptext.com/articles/aleph-number).\nThe aleph numbers are indexed by [ordinal numbers](https://bliptext.com/articles/ordinal-number).\nIf the [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is true, this transfinite sequence includes every cardinal number.\nIf the axiom of choice is not true (see ), there are infinite cardinals that are not aleph numbers.\n[Cardinality](https://bliptext.com/articles/cardinality) is studied for its own sake as part of [set theory](https://bliptext.com/articles/set-theory).\nIt is also a tool used in branches of mathematics including [model theory](https://bliptext.com/articles/model-theory), [combinatorics](https://bliptext.com/articles/combinatorics), [abstract algebra](https://bliptext.com/articles/abstract-algebra) and [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIn [category theory](https://bliptext.com/articles/category-theory), the cardinal numbers form a [skeleton](https://bliptext.com/articles/skeleton-category-theory) of the [category of sets](https://bliptext.com/articles/category-of-sets).\n# [A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. | upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg]\n# [Aleph-null, the smallest infinite cardinal | upload.wikimedia.org/wikipedia/commons/e/e0/Aleph0.svg]\n\n# History\nThe notion of cardinality, as now understood, was formulated by [Georg Cantor](https://bliptext.com/articles/georg-cantor), the originator of [set theory](https://bliptext.com/articles/set-theory), in 1874–1884.\nCardinality can be used to compare an aspect of finite sets.\nFor example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three.\nThis is established by the existence of a [bijection](https://bliptext.com/articles/bijection) (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.\nCantor applied his concept of bijection to infinite sets (for example the set of natural numbers **N** = {0, 1, 2, 3, ...}).\nThus, he called all sets having a bijection with **N** denumerable (countably infinite) sets, which all share the same cardinal number.\nThis cardinal number is called 4 null.\nHe called the cardinal numbers of infinite sets [transfinite cardinal numbers](https://bliptext.com/articles/transfinite-cardinal-numbers).\nCantor proved that any [unbounded subset](https://bliptext.com/articles/bounded-set) of ****N**** has the same cardinality as ****N****, even though this might appear to run contrary to intuition.\nHe also proved that the set of all [ordered pairs](https://bliptext.com/articles/ordered-pair) of natural numbers is denumerable; this implies that the set of all [rational numbers](https://bliptext.com/articles/rational-number) is also denumerable, since every rational can be represented by a pair of integers.\nHe later proved that the set of all real [algebraic numbers](https://bliptext.com/articles/algebraic-number) is also denumerable.\nEach real algebraic number z may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai ∈ **Z** together with a pair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies in the interval (b0, b1).\nIn his 1874 paper \"On a Property of the Collection of All Real Algebraic **N**umbers\", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of **N**.\nHis proof used an argument with [nested intervals](https://bliptext.com/articles/nested-intervals), but in an 1891 paper, he proved the same result using his ingenious and much simpler [diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument).\nThe new cardinal number of the set of real numbers is called the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) and Cantor used the symbol for it.\nCantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (\\aleph_0, aleph-null), and that for every cardinal number there is a next-larger cardinal\nHis [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) is the proposition that the cardinality of the set of real numbers is the same as \\aleph_1.\nThis hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them.\nThis was shown in 1963 by [Paul Cohen](https://bliptext.com/articles/paul-cohen-mathematician), complementing earlier work by [Kurt Gödel](https://bliptext.com/articles/kurt-g-del) in 1940.\n\n# Motivation\nIn informal use, a cardinal number is what is normally referred to as a [counting number](https://bliptext.com/articles/counting-number), provided that 0 is included: 0, 1, 2, .... They may be identified with the [natural numbers](https://bliptext.com/articles/natural-numbers) beginning with 0.\nThe counting numbers are exactly what can be defined formally as the [finite](https://bliptext.com/articles/finite-set) cardinal numbers.\nInfinite cardinals only occur in higher-level mathematics and [logic](https://bliptext.com/articles/logic).\nMore formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence.\nFor finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size.\nFor example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.\nHowever, when dealing with [infinite sets](https://bliptext.com/articles/infinite-set), it is essential to distinguish between the two, since the two notions are in fact different for [infinite sets](https://bliptext.com/articles/infinite-set).\nConsidering the position aspect leads to [ordinal numbers](https://bliptext.com/articles/ordinal-numbers), while the size aspect is generalized by the cardinal numbers described here.\nThe intuition behind the formal definition of cardinal is the construction of a notion of the relative size or \"bigness\" of a set, without reference to the kind of members which it has.\nFor finite sets this is easy; one simply counts the number of elements a set has.\nIn order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.\nA set Y is at least as big as a set X if there is an [injective](https://bliptext.com/articles/injective-function) [mapping](https://bliptext.com/articles/map-mathematics) from the elements of X to the elements of Y.\nAn injective mapping identifies each element of the set X with a unique element of the set Y.\nThis is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:\nwhich is injective, and hence conclude that Y has cardinality greater than or equal to X.\nThe element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a [bijective](https://bliptext.com/articles/bijective) mapping.\nThe advantage of this notion is that it can be extended to infinite sets.\nWe can then extend this to an equality-style relation.\nTwo [sets](https://bliptext.com/articles/set-mathematics) X and Y are said to have the same cardinality if there exists a [bijection](https://bliptext.com/articles/bijection) between X and Y.\nBy the [Schroeder–Bernstein theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem), this is equivalent to there being both an injective mapping from X to Y, and an injective mapping from Y to X.\nWe then write |X| = |Y|.\nThe cardinal number of X itself is often defined as the least ordinal a with |a| = |X|.\nThis is called the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment); for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the [well-ordering principle](https://bliptext.com/articles/well-ordering-principle).\nIt is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.\nThe classic example used is that of the infinite hotel paradox, also called [Hilbert's paradox of the Grand Hotel](https://bliptext.com/articles/hilbert-s-paradox-of-the-grand-hotel).\nSupposing there is an innkeeper at a hotel with an infinite number of rooms.\nThe hotel is full, and then a new guest arrives.\nIt is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant.\nWe can explicitly write a segment of this mapping:\nWith this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown.\nThis motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a [Dedekind-infinite set](https://bliptext.com/articles/dedekind-infinite-set)); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.\nWhen considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets.\nIt happens that it does not; by considering the above example we can see that if some object \"one greater than infinity\" exists, then it must have the same cardinality as the infinite set we started out with.\nIt is possible to use a different formal notion for number, called [ordinals](https://bliptext.com/articles/ordinal-number), based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.\nIt can be proved that the cardinality of the [real numbers](https://bliptext.com/articles/real-number) is greater than that of the natural numbers just described.\nThis can be visualized using [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument); classic questions of cardinality (for instance the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis)) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.\nIn more recent times, mathematicians have been describing the properties of larger and larger cardinals.\nSince cardinality is such a common concept in mathematics, a variety of names are in use.\nSameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.\nIt is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.\n\n# Formal definition\nFormally, assuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), the cardinality of a set X is the least [ordinal number](https://bliptext.com/articles/ordinal-number) α such that there is a bijection between X and α.\nThis definition is known as the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment).\nIf the axiom of choice is not assumed, then a different approach is needed.\nThe oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and [Principia Mathematica](https://bliptext.com/articles/principia-mathematica)) is as the class [X] of all sets that are equinumerous with X.\nThis does not work in [ZFC](https://bliptext.com/articles/zfc) or other related systems of [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory) because if X is non-empty, this collection is too large to be a set.\nIn fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X, and so by the [axiom of limitation of size](https://bliptext.com/articles/axiom-of-limitation-of-size), [X] is a proper class.\nThe definition does work however in [type theory](https://bliptext.com/articles/type-theory) and in [New Foundations](https://bliptext.com/articles/new-foundations) and related systems.\nHowever, if we restrict from this class to those equinumerous with X that have the least [rank](https://bliptext.com/articles/rank-set-theory), then it will work (this is a trick due to [Dana Scott](https://bliptext.com/articles/dana-scott): it works because the collection of objects with any given [rank](https://bliptext.com/articles/rank-set-theory) is a set).\nVon Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers.\nHowever, they differ for infinite numbers.\nFor example, in ordinal arithmetic while in cardinal arithmetic, although the von Neumann assignment puts.\nOn the other hand, Scott's trick implies that the cardinal number 0 is, which is also the ordinal number 1, and this may be confusing.\nA possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.\nFormally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an [injective](https://bliptext.com/articles/injective) function from X to Y.\nThe [Cantor–Bernstein–Schroeder theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem) states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|.\nThe axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|.\nA set X is [Dedekind-infinite](https://bliptext.com/articles/dedekind-infinite) if there exists a [proper subset](https://bliptext.com/articles/proper-subset) Y of X with |X| = |Y|, and [Dedekind-finite](https://bliptext.com/articles/dedekind-finite) if such a subset does not exist.\nThe [finite](https://bliptext.com/articles/finite-set) cardinals are just the [natural numbers](https://bliptext.com/articles/natural-numbers), in the sense that a set X is [finite](https://bliptext.com/articles/finite-set) if and only if |X| = |n| = n for some natural number n.\nAny other set is [infinite](https://bliptext.com/articles/infinite-set).\nAssuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones.\nIt can also be proved that the cardinal \\aleph_0 ([aleph null](https://bliptext.com/articles/aleph-null) or aleph-0, where aleph is the first letter in the [Hebrew alphabet](https://bliptext.com/articles/hebrew-alphabet), represented \\aleph) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality \\aleph_0).\nThe next larger cardinal is denoted by \\aleph_1, and so on.\nFor every ordinal α, there is a cardinal number and this list exhausts all infinite cardinal numbers.\n\n# Cardinal arithmetic\nWe can define [arithmetic](https://bliptext.com/articles/arithmetic) operations on cardinal numbers that generalize the ordinary operations for natural numbers.\nIt can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers.\nFurthermore, these operations share many properties with ordinary arithmetic.\n\n## Successor cardinal\nIf the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor.\n(Without the axiom of choice, using [Hartogs' theorem](https://bliptext.com/articles/hartogs-number), it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that ) For finite cardinals, the successor is simply κ + 1.\nFor infinite cardinals, the successor cardinal differs from the [successor ordinal](https://bliptext.com/articles/successor-ordinal).\n\n## Cardinal addition\nIf X and Y are [disjoint](https://bliptext.com/articles/disjoint-sets), addition is given by the [union](https://bliptext.com/articles/union-set-theory) of X and Y.\nIf the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).\nZero is an additive identity κ + 0 = 0 + κ = κ.\nAddition is [associative](https://bliptext.com/articles/associative) (κ + μ) + ν = κ + (μ + ν).\nAddition is [commutative](https://bliptext.com/articles/commutative) κ + μ = μ + κ.\nAddition is non-decreasing in both arguments:\nAssuming the axiom of choice, addition of infinite cardinal numbers is easy.\nIf either κ or μ is infinite, then\n\n### Subtraction\nAssuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ.\nIt will be unique (and equal to σ) if and only if μ < σ.\n\n## Cardinal multiplication\nThe product of cardinals comes from the [Cartesian product](https://bliptext.com/articles/cartesian-product).\nκ·0 = 0·κ = 0.\nκ·μ = 0 → (κ = 0 or μ = 0).\nOne is a multiplicative identity κ·1 = 1·κ = κ.\nMultiplication is associative (κ·μ)·ν = κ·(μ·ν).\nMultiplication is [commutative](https://bliptext.com/articles/commutative) κ·μ = μ·κ.\nMultiplication is non-decreasing in both arguments:\nκ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).\nMultiplication [distributes](https://bliptext.com/articles/distributivity) over addition:\nκ·(μ + ν) = κ·μ + κ·ν and\n(μ + ν)·κ = μ·κ + ν·κ.\nAssuming the axiom of choice, multiplication of infinite cardinal numbers is also easy.\nIf either κ or μ is infinite and both are non-zero, then\n\n### Division\nAssuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π.\nIt will be unique (and equal to π) if and only if μ < π.\n\n## Cardinal exponentiation\nExponentiation is given by where XY is the set of all [functions](https://bliptext.com/articles/function-mathematics) from Y to X.\nIt is easy to check that the right-hand side depends only on {|X|} and {|Y|}.\nExponentiation is non-decreasing in both arguments:\n2undefined is the cardinality of the [power set](https://bliptext.com/articles/power-set) of the set X and [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument) shows that 2undefined > |X| for any set X.\nThis proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ).\nIn fact, the [class](https://bliptext.com/articles/class-set-theory) of cardinals is a proper [class](https://bliptext.com/articles/class-set-theory).\n(This proof fails in some set theories, notably [New Foundations](https://bliptext.com/articles/new-foundations).)\nAll the remaining propositions in this section assume the axiom of choice:\nIf 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:\nUsing [König's theorem](https://bliptext.com/articles/k-nig-s-theorem-set-theory), one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the [cofinality](https://bliptext.com/articles/cofinality) of κ.\n\n### Roots\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying will be \\kappa.\n\n### Logarithms\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying.\nHowever, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy.\nThe logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ.\nLogarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of [cardinal invariants](https://bliptext.com/articles/cardinal-invariant) of [topological spaces](https://bliptext.com/articles/topological-space), though they lack some of the properties that logarithms of positive real numbers possess.\n\n# The continuum hypothesis\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) (CH) states that there are no cardinals strictly between \\aleph_0 and The latter cardinal number is also often denoted by ; it is the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) (the set of [real numbers](https://bliptext.com/articles/real-number)).\nIn this case\nSimilarly, the [generalized continuum hypothesis](https://bliptext.com/articles/generalized-continuum-hypothesis) (GCH) states that for every infinite cardinal \\kappa, there are no cardinals strictly between \\kappa and 2^\\kappa.\nBoth the continuum hypothesis and the generalized continuum hypothesis have been proved to be independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ([ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory)).\nIndeed, [Easton's theorem](https://bliptext.com/articles/easton-s-theorem) shows that, for [regular cardinals](https://bliptext.com/articles/regular-cardinal) \\kappa, the only restrictions ZFC places on the cardinality of 2^\\kappa are that, and that the exponential function is non-decreasing.","current_revision":"0caabb7f-df6d-45bb-8547-f0d74c224997","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:32:25.570Z","updated_at":"2026-01-28T10:23:23.787Z","search_vector":"'cardin':1 'number':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a ****cardinal** number**, or **cardinal** for short, is what is commonly called the number of elements of a [set](https://bliptext.com/articles/set-mathematics).\nIn the case of a [finite set](https://bliptext.com/articles/finite-set), its cardinal number, or [cardinality](https://bliptext.com/articles/cardinality) is therefore a [natural number](https://bliptext.com/articles/natural-number).\nFor dealing with the case of [infinite sets](https://bliptext.com/articles/infinite-set), the [infinite cardinal numbers](https://bliptext.com/articles/transfinite-number) have been introduced, which are often denoted with the [Hebrew letter](https://bliptext.com/articles/hebrew-alphabet) \\[aleph](https://bliptext.com/articles/aleph-hebrew) ([aleph](https://bliptext.com/articles/aleph-hebrew)) marked with subscript indicating their rank among the infinite cardinals.\nCardinality is defined in terms of [bijective functions](https://bliptext.com/articles/bijective-function).\nTwo sets have the same cardinality [if, and only if](https://bliptext.com/articles/if-and-only-if), there is a [one-to-one correspondence](https://bliptext.com/articles/one-to-one-correspondence) (bijection) between the elements of the two sets.\nIn the case of finite sets, this agrees with the intuitive notion of number of elements.\nIn the case of infinite sets, the behavior is more complex.\nA fundamental theorem due to [Georg Cantor](https://bliptext.com/articles/georg-cantor) shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [real numbers](https://bliptext.com/articles/real-number) is greater than the cardinality of the set of natural numbers.\nIt is also possible for a [proper subset](https://bliptext.com/articles/proper-subset) of an infinite set to have the same cardinality as the original set—something that cannot happen with [proper subset](https://bliptext.com/articles/proper-subset)s of finite sets.\nThere is a [transfinite sequence](https://bliptext.com/articles/transfinite-sequence) of cardinal numbers:\nThis sequence starts with the [natural numbers](https://bliptext.com/articles/natural-number) including zero (finite cardinals), which are followed by the [aleph numbers](https://bliptext.com/articles/aleph-number).\nThe aleph numbers are indexed by [ordinal numbers](https://bliptext.com/articles/ordinal-number).\nIf the [axiom of choice](https://bliptext.com/articles/axiom-of-choice) is true, this transfinite sequence includes every cardinal number.\nIf the axiom of choice is not true (see ), there are infinite cardinals that are not aleph numbers.\n[Cardinality](https://bliptext.com/articles/cardinality) is studied for its own sake as part of [set theory](https://bliptext.com/articles/set-theory).\nIt is also a tool used in branches of mathematics including [model theory](https://bliptext.com/articles/model-theory), [combinatorics](https://bliptext.com/articles/combinatorics), [abstract algebra](https://bliptext.com/articles/abstract-algebra) and [mathematical analysis](https://bliptext.com/articles/mathematical-analysis).\nIn [category theory](https://bliptext.com/articles/category-theory), the cardinal numbers form a [skeleton](https://bliptext.com/articles/skeleton-category-theory) of the [category of sets](https://bliptext.com/articles/category-of-sets).\n# [A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. | upload.wikimedia.org/wikipedia/commons/a/a5/Bijection.svg]\n# [Aleph-null, the smallest infinite cardinal | upload.wikimedia.org/wikipedia/commons/e/e0/Aleph0.svg]\n\n# History\nThe notion of cardinality, as now understood, was formulated by [Georg Cantor](https://bliptext.com/articles/georg-cantor), the originator of [set theory](https://bliptext.com/articles/set-theory), in 1874–1884.\nCardinality can be used to compare an aspect of finite sets.\nFor example, the sets {1,2,3} and {4,5,6} are not equal, but have the same cardinality, namely three.\nThis is established by the existence of a [bijection](https://bliptext.com/articles/bijection) (i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.\nCantor applied his concept of bijection to infinite sets (for example the set of natural numbers **N** = {0, 1, 2, 3, ...}).\nThus, he called all sets having a bijection with **N** denumerable (countably infinite) sets, which all share the same cardinal number.\nThis cardinal number is called \\aleph_0, [aleph-null](https://bliptext.com/articles/aleph-number).\nHe called the cardinal numbers of infinite sets [transfinite cardinal numbers](https://bliptext.com/articles/transfinite-cardinal-numbers).\nCantor proved that any [unbounded subset](https://bliptext.com/articles/bounded-set) of ****N**** has the same cardinality as ****N****, even though this might appear to run contrary to intuition.\nHe also proved that the set of all [ordered pairs](https://bliptext.com/articles/ordered-pair) of natural numbers is denumerable; this implies that the set of all [rational numbers](https://bliptext.com/articles/rational-number) is also denumerable, since every rational can be represented by a pair of integers.\nHe later proved that the set of all real [algebraic numbers](https://bliptext.com/articles/algebraic-number) is also denumerable.\nEach real algebraic number z may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (a0, a1, ..., an), ai ∈ **Z** together with a pair of rationals (b0, b1) such that z is the unique root of the polynomial with coefficients (a0, a1, ..., an) that lies in the interval (b0, b1).\nIn his 1874 paper \"On a Property of the Collection of All Real Algebraic **N**umbers\", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of **N**.\nHis proof used an argument with [nested intervals](https://bliptext.com/articles/nested-intervals), but in an 1891 paper, he proved the same result using his ingenious and much simpler [diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument).\nThe new cardinal number of the set of real numbers is called the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) and Cantor used the symbol for it.\nCantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (\\aleph_0, aleph-null), and that for every cardinal number there is a next-larger cardinal\nHis [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) is the proposition that the cardinality of the set of real numbers is the same as \\aleph_1.\nThis hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them.\nThis was shown in 1963 by [Paul Cohen](https://bliptext.com/articles/paul-cohen-mathematician), complementing earlier work by [Kurt Gödel](https://bliptext.com/articles/kurt-g-del) in 1940.\n\n# Motivation\nIn informal use, a cardinal number is what is normally referred to as a [counting number](https://bliptext.com/articles/counting-number), provided that 0 is included: 0, 1, 2, .... They may be identified with the [natural numbers](https://bliptext.com/articles/natural-numbers) beginning with 0.\nThe counting numbers are exactly what can be defined formally as the [finite](https://bliptext.com/articles/finite-set) cardinal numbers.\nInfinite cardinals only occur in higher-level mathematics and [logic](https://bliptext.com/articles/logic).\nMore formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence.\nFor finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size.\nFor example, 3 describes the position of 'c' in the sequence , and we can construct the set {a,b,c}, which has 3 elements.\nHowever, when dealing with [infinite sets](https://bliptext.com/articles/infinite-set), it is essential to distinguish between the two, since the two notions are in fact different for [infinite sets](https://bliptext.com/articles/infinite-set).\nConsidering the position aspect leads to [ordinal numbers](https://bliptext.com/articles/ordinal-numbers), while the size aspect is generalized by the cardinal numbers described here.\nThe intuition behind the formal definition of cardinal is the construction of a notion of the relative size or \"bigness\" of a set, without reference to the kind of members which it has.\nFor finite sets this is easy; one simply counts the number of elements a set has.\nIn order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.\nA set Y is at least as big as a set X if there is an [injective](https://bliptext.com/articles/injective-function) [mapping](https://bliptext.com/articles/map-mathematics) from the elements of X to the elements of Y.\nAn injective mapping identifies each element of the set X with a unique element of the set Y.\nThis is most easily understood by an example; suppose we have the sets X = {1,2,3} and Y = {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:\nwhich is injective, and hence conclude that Y has cardinality greater than or equal to X.\nThe element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a [bijective](https://bliptext.com/articles/bijective) mapping.\nThe advantage of this notion is that it can be extended to infinite sets.\nWe can then extend this to an equality-style relation.\nTwo [sets](https://bliptext.com/articles/set-mathematics) X and Y are said to have the same cardinality if there exists a [bijection](https://bliptext.com/articles/bijection) between X and Y.\nBy the [Schroeder–Bernstein theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem), this is equivalent to there being both an injective mapping from X to Y, and an injective mapping from Y to X.\nWe then write |X| = |Y|.\nThe cardinal number of X itself is often defined as the least ordinal a with |a| = |X|.\nThis is called the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment); for this definition to make sense, it must be proved that every set has the same cardinality as some ordinal; this statement is the [well-ordering principle](https://bliptext.com/articles/well-ordering-principle).\nIt is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.\nThe classic example used is that of the infinite hotel paradox, also called [Hilbert's paradox of the Grand Hotel](https://bliptext.com/articles/hilbert-s-paradox-of-the-grand-hotel).\nSupposing there is an innkeeper at a hotel with an infinite number of rooms.\nThe hotel is full, and then a new guest arrives.\nIt is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant.\nWe can explicitly write a segment of this mapping:\nWith this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown.\nThis motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a [Dedekind-infinite set](https://bliptext.com/articles/dedekind-infinite-set)); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.\nWhen considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets.\nIt happens that it does not; by considering the above example we can see that if some object \"one greater than infinity\" exists, then it must have the same cardinality as the infinite set we started out with.\nIt is possible to use a different formal notion for number, called [ordinals](https://bliptext.com/articles/ordinal-number), based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.\nIt can be proved that the cardinality of the [real numbers](https://bliptext.com/articles/real-number) is greater than that of the natural numbers just described.\nThis can be visualized using [Cantor's diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument); classic questions of cardinality (for instance the [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis)) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.\nIn more recent times, mathematicians have been describing the properties of larger and larger cardinals.\nSince cardinality is such a common concept in mathematics, a variety of names are in use.\nSameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.\nIt is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.\n\n# Formal definition\nFormally, assuming the [axiom of choice](https://bliptext.com/articles/axiom-of-choice), the cardinality of a set X is the least [ordinal number](https://bliptext.com/articles/ordinal-number) α such that there is a bijection between X and α.\nThis definition is known as the [von Neumann cardinal assignment](https://bliptext.com/articles/von-neumann-cardinal-assignment).\nIf the axiom of choice is not assumed, then a different approach is needed.\nThe oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and [Principia Mathematica](https://bliptext.com/articles/principia-mathematica)) is as the class [X] of all sets that are equinumerous with X.\nThis does not work in [ZFC](https://bliptext.com/articles/zfc) or other related systems of [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory) because if X is non-empty, this collection is too large to be a set.\nIn fact, for X ≠ ∅ there is an injection from the universe into [X] by mapping a set m to {m} × X, and so by the [axiom of limitation of size](https://bliptext.com/articles/axiom-of-limitation-of-size), [X] is a proper class.\nThe definition does work however in [type theory](https://bliptext.com/articles/type-theory) and in [New Foundations](https://bliptext.com/articles/new-foundations) and related systems.\nHowever, if we restrict from this class to those equinumerous with X that have the least [rank](https://bliptext.com/articles/rank-set-theory), then it will work (this is a trick due to [Dana Scott](https://bliptext.com/articles/dana-scott): it works because the collection of objects with any given [rank](https://bliptext.com/articles/rank-set-theory) is a set).\nVon Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers.\nHowever, they differ for infinite numbers.\nFor example, in ordinal arithmetic while in cardinal arithmetic, although the von Neumann assignment puts.\nOn the other hand, Scott's trick implies that the cardinal number 0 is, which is also the ordinal number 1, and this may be confusing.\nA possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.\nFormally, the order among cardinal numbers is defined as follows: |X| ≤ |Y| means that there exists an [injective](https://bliptext.com/articles/injective) function from X to Y.\nThe [Cantor–Bernstein–Schroeder theorem](https://bliptext.com/articles/cantor-bernstein-schroeder-theorem) states that if |X| ≤ |Y| and |Y| ≤ |X| then |X| = |Y|.\nThe axiom of choice is equivalent to the statement that given two sets X and Y, either |X| ≤ |Y| or |Y| ≤ |X|.\nA set X is [Dedekind-infinite](https://bliptext.com/articles/dedekind-infinite) if there exists a [proper subset](https://bliptext.com/articles/proper-subset) Y of X with |X| = |Y|, and [Dedekind-finite](https://bliptext.com/articles/dedekind-finite) if such a subset does not exist.\nThe [finite](https://bliptext.com/articles/finite-set) cardinals are just the [natural numbers](https://bliptext.com/articles/natural-numbers), in the sense that a set X is [finite](https://bliptext.com/articles/finite-set) if and only if |X| = |n| = n for some natural number n.\nAny other set is [infinite](https://bliptext.com/articles/infinite-set).\nAssuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones.\nIt can also be proved that the cardinal \\aleph_0 ([aleph null](https://bliptext.com/articles/aleph-null) or aleph-0, where aleph is the first letter in the [Hebrew alphabet](https://bliptext.com/articles/hebrew-alphabet), represented \\aleph) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality \\aleph_0).\nThe next larger cardinal is denoted by \\aleph_1, and so on.\nFor every ordinal α, there is a cardinal number and this list exhausts all infinite cardinal numbers.\n\n# Cardinal arithmetic\nWe can define [arithmetic](https://bliptext.com/articles/arithmetic) operations on cardinal numbers that generalize the ordinary operations for natural numbers.\nIt can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers.\nFurthermore, these operations share many properties with ordinary arithmetic.\n\n## Successor cardinal\nIf the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor.\n(Without the axiom of choice, using [Hartogs' theorem](https://bliptext.com/articles/hartogs-number), it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that ) For finite cardinals, the successor is simply κ + 1.\nFor infinite cardinals, the successor cardinal differs from the [successor ordinal](https://bliptext.com/articles/successor-ordinal).\n\n## Cardinal addition\nIf X and Y are [disjoint](https://bliptext.com/articles/disjoint-sets), addition is given by the [union](https://bliptext.com/articles/union-set-theory) of X and Y.\nIf the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).\nZero is an additive identity κ + 0 = 0 + κ = κ.\nAddition is [associative](https://bliptext.com/articles/associative) (κ + μ) + ν = κ + (μ + ν).\nAddition is [commutative](https://bliptext.com/articles/commutative) κ + μ = μ + κ.\nAddition is non-decreasing in both arguments:\nAssuming the axiom of choice, addition of infinite cardinal numbers is easy.\nIf either κ or μ is infinite, then\n\n### Subtraction\nAssuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ.\nIt will be unique (and equal to σ) if and only if μ |X| for any set X.\nThis proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2κ).\nIn fact, the [class](https://bliptext.com/articles/class-set-theory) of cardinals is a proper [class](https://bliptext.com/articles/class-set-theory).\n(This proof fails in some set theories, notably [New Foundations](https://bliptext.com/articles/new-foundations).)\nAll the remaining propositions in this section assume the axiom of choice:\nIf 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:\nUsing [König's theorem](https://bliptext.com/articles/k-nig-s-theorem-set-theory), one can prove κ < κcf(κ) and κ < cf(2κ) for any infinite cardinal κ, where cf(κ) is the [cofinality](https://bliptext.com/articles/cofinality) of κ.\n\n### Roots\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying will be \\kappa.\n\n### Logarithms\nAssuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying.\nHowever, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy.\nThe logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ.\nLogarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of [cardinal invariants](https://bliptext.com/articles/cardinal-invariant) of [topological spaces](https://bliptext.com/articles/topological-space), though they lack some of the properties that logarithms of positive real numbers possess.\n\n# The continuum hypothesis\nThe [continuum hypothesis](https://bliptext.com/articles/continuum-hypothesis) (CH) states that there are no cardinals strictly between \\aleph_0 and The latter cardinal number is also often denoted by ; it is the [cardinality of the continuum](https://bliptext.com/articles/cardinality-of-the-continuum) (the set of [real numbers](https://bliptext.com/articles/real-number)).\nIn this case\nSimilarly, the [generalized continuum hypothesis](https://bliptext.com/articles/generalized-continuum-hypothesis) (GCH) states that for every infinite cardinal \\kappa, there are no cardinals strictly between \\kappa and 2^\\kappa.\nBoth the continuum hypothesis and the generalized continuum hypothesis have been proved to be independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ([ZFC](https://bliptext.com/articles/zermelo-fraenkel-set-theory)).\nIndeed, [Easton's theorem](https://bliptext.com/articles/easton-s-theorem) shows that, for [regular cardinals](https://bliptext.com/articles/regular-cardinal) \\kappa, the only restrictions ZFC places on the cardinality of 2^\\kappa are that, and that the exponential function is non-decreasing.","revisionCount":2},"hasMore":false,"nextCursor":null}